deletions | additions       diff --git a/Methodolofy Measurement Definitions.tex b/Methodolofy Measurement Definitions.tex  index c28ca84..66b0709 100644  --- a/Methodolofy Measurement Definitions.tex  +++ b/Methodolofy Measurement Definitions.tex 
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\subsubsection{Interactions}  An interaction quantifies the spatial relationship between moving objects in a scene. At the most fundamental level, an interaction is defined as a pair of moving objects simultaneously present in a scene over a common time interval (also reffered referred  to as a user pair). We further define an instantaneous observation (i.e. in a give video frame) within this time interval as an interaction-instant. This interaction definition is generic, if not naive, as the quality depends largely on how the scene is constructed. For example, the significance of an interaction between two vehicles seperated separated  from each other physically (e.g. via a median or a large building) may not be comparable to an interaction between two vehicles merely seperated separated  by a painted line because the probability that one of the vehicles comes into contact with the other vehicle is reduced in the case of the median. This may interfere with collision prediction attempts, particularly if scenes are not consistently selected and geometry is not controlled. One solution is to perform a triage of user pairs based on physical access and proximity. A network topology coupled with a driving distance horizon is proposed. This is not a perfect solution, however, as physical access isn't necessarily a binary option. In our median example, it is still physically possible, although less likely, for vehicles to cross-over into an opposing lane and cause a collision, although this is something that could be modeled. modelled.  \subsubsection{Motion Prediction}  \label{motion-prediction}  While vehicle trajectories offer a rich set of observed behavioural data, they do not provide much collision data; this is by design of the proactive road safety approach: predicting collisions should be performed without observing them directly. In order to study collisions, they need to be extrapolated from traffic events with potential for collision. This potential is modeled modelled  by predicting future positions of vehicles using motion prediction at every instant in time and examining i) situations of particular probability of collision (i.e. threshold) or ii) evolution of the probability of collision over a timeseries. time series.  Several motion prediction models are proposed for study \cite{Mohamed_2013}: \begin{itemize}  \item \textbf{Constant velocity} is the classic motion prediction model, wherein vehicles are projected along straight paths at a constant speed and heading using the velocity vector at that moment in time. This model is the simplest but also makes the most assumptions: only one movement is predicted at every instant, both users do not enter evasive action in the event of a collision course, and the natural (non-reacting) motion of a moving object is a straight path (not always true). These assumptions may be adequate for specific applications of the methodology, e.g. highways \cite{St_Aubin_2013}. The current implementation is based off of \cite{Laureshyn_2010}.  \item \textbf{Normal adaptation} uses the initial velocity vector at the prediction moment to project trajectories, but modifies the velocity vector to account for normal variation. This model benefits from a wider range of possible outcome velocity vectors, but otherwise suffers the same problems and makes the same assumptions as constant velocity. The implementation of normal adaption adaptation  studied is based off of \cite{Mohamed_2013}, using a acceleration maximum $\alpha$ of \begin{equation} \label{eqn:norm-adapt-accel-maxima} \alpha = \pm \frac{2}{f^2} \end{equation}  and a maximum steering paramter parameter  $\sigma$ of \begin{equation} \label{eqn:norm-adapt-steering-maxima} \sigma = \frac{0.2}{f} \end{equation} 
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\item \textbf{Motion patterns} are a family of models which use machine learning to calculate future position likelihoods from past behaviour \cite{saunier07probabilistic}. This type of model is the most promising as motion prediction is probabilistic in nature and inherently models naturalistic behaviour. However, they may not be able to model erratic behaviour such as roadway departures. Motion patterns are also complex to implement and expensive to process. The type of motion pattern being studied for implementation is a discretised motion pattern \cite{St_Aubin_2014}.  \end{itemize}  As illustrated in Figure~\ref{fig:prob-collision-space}, motion prediction is performed for each user pair over each interaction-instant $t_0$ for a number of timesteps time steps  of size $\Delta t$ between $t_0$ and some chosen timehorizon. Each motion prediction may generate for two road users a series or a matrix of collision points with a sum of probabilities inferior or equal to 1. This is significantly larger and more difficult to handle than trajectory data, and currently cannot be performed in real time. A sample analysis for a single pair of road users at a single site over a timeseries time series  of 64 interaction-instants, lasting just over 4 seconds is presented in Figure~\ref{fig:conflict-video} and Figure~\ref{fig:conflict-series}. In this scenario, vehicle # 304 is approaching at high velocity vehicle # 303 which is enganged engaged  in an illegal u-turn U-turn  (in a right-hand roundabout, users are supposed to travel counter-clockwise around the center centre  island at all times). The differential velocity $\Delta v$, realtive relative  distance $d$, and corresponding time $t$ is measured for every interaction-instant. In a matter of just under 4 seconds, the differential velocity changes from 9.63 m/s to 2.26 while the relative distance changes from 28.57 metres to 9.57 metres. For every interation-instant interaction-instant  of this user pair, motion prediction is used to calculate resulting TTC under each motion prediction's respective hypothesis. These predicted collisions and associated TTC measures are presented in Figure~\ref{fig:ttc-timeseries}. Motion pattern prediction generates many more possible collision points than constant velocity prediction, though each of these points has a significantly lower associated probability. For beterr better  instant-to-instant comparison, a summary $TTC'_i$ at time $t_i$ is calculated as the probability-weighted TTC average \begin{equation} \label{eqn:TTC-mp-weight-avg} TTC'_i = \frac{\sum_{j=1}^{n}{TTC_{ij} Prob(collision)_{ij}} }{n} \end{equation}